Expander Graphs and Property (t)
نویسنده
چکیده
Families of expander graphs are sparse graphs such that the number of vertices in each graph grows yet each graph remains difficult to disconnect. Expander graphs are of great importance in theoretical computer science. In this paper we study the connection between the Cheeger constant, a measure of the connectivity of the graph, and the smallest nonzero eigenvalue of the graph Laplacian. We show for expander graphs these two numbers are strictly bounded away from zero. Given a finitely generated locally compact group satisfying Kazhdan’s property (T), we construct expanders from the Cayley graphs of finite index normal subgroups with finite generating sets. We follow Alexander Lubotzky’s treatment in [7].
منابع مشابه
Pcmi Lecture Notes on Property (t ), Expander Graphs and Approximate Groups (preliminary Version)
The final aim of these lectures will be to prove spectral gaps for finite groups and to turn certain Cayley graphs into expander graphs. However in order to do so it is useful to have some understanding of the analogous spectral notions of amenability and Kazhdan property (T ) which are important for infinite groups. In fact one important aspect of asymptotic group theory (the part of group the...
متن کاملKolmogorov-barzdin and Spacial Realizations of Expander Graphs
One application of graph theory is to analyze connectivity of neurons and axons in the brain. We begin with basic definitions from graph theory including the Cheeger constant, a measure of connectivity of a graph. In Section 2, we will examine expander graphs, which are very sparse yet highly connected. Surprisingly, not only do expander graphs exist, but most random graphs have the expander pr...
متن کاملGeometric Property ( T )
This paper discusses ‘geometric property (T)’. This is a property of metric spaces introduced in earlier work of the authors for its applications to K-theory. Geometric property (T) is a strong form of ‘expansion property’: in particular for a sequence (Xn) of bounded degree finite graphs, it is strictly stronger than (Xn) being an expander in the sense that the Cheeger constants h(Xn) are boun...
متن کاملSymmetric Groups and Expanders
We construct an explicit generating sets Fn and F̃n of the alternating and the symmetric groups, which make the Cayley graphs C(Alt(n), Fn) and C(Sym(n), F̃n) a family of bounded degree expanders for all sufficiently large n. These expanders have many applications in the theory of random walks on groups and other areas of mathematics. A finite graph Γ is called an ǫ-expander for some ǫ ∈ (0, 1), ...
متن کاملLecture 4: Approximate Groups and the Bourgain-gamburd Method (preliminary Version)
Up until the Bourgain-Gamburd 2005 breakthrough the only known ways to turn SLd(Fp) into an expander graph (i.e. to find a generating set of small size whose associated Cayley graph has a good spectral gap) was either through property (T ) (as in the Margulis construction) when d > 3 or through the Selberg property (and the dictionary between combinatorial expansion of the Cayley graphs and the...
متن کامل